Abstract
We present a numerical technique for the computation of a Lyapunov functionfor nonlinear systems with an asymptotically stable equilibrium point. The proposed approachconstructs a partition of the state space, called a triangulation, and then computes values atthe vertices of the triangulation using a Lyapunov function from a classical converse Lyapunovtheorem due to Yoshizawa. A simple interpolation of the vertex values then yields a Continuousand Piecewise Affine (CPA) function.Verification that the obtained CPA function is a Lyapunov function is shownto be equivalent to verification of several simple inequalities.Numerical examples are presented demonstrating different aspects of the proposed method.
Highlights
Lyapunov’s Second or Direct Method [21] has proved to be one of the most useful tools for demonstrating stability properties
For a suitable triangulation of the state space, at each simplex vertex we calculate the value of a Lyapunov function construction due to Yoshizawa [29, 30]
We can verify that the continuous and piecewise affine (CPA) function defined is a Lyapunov function (Corollary 1) by checking a simple linear inequality (7) at each vertex of the triangulation
Summary
Lyapunov’s Second or Direct Method [21] (see [14, 26, 30]) has proved to be one of the most useful tools for demonstrating stability properties. In our approach, this finite time solution is not required for every initial condition in the considered region, but only at the vertices of the triangulation It is satisfaction of the aforementioned linear inequalities that is the crucial step in demonstrating a CPA Lyapunov function rather than constructing a numerical approximation of the construction of Yoshizawa. We may wish to use very small simplices in order to reduce the error between a given Lyapunov function and its CPA approximation, and a reasonable measure of distance-to-degeneracy should scale the spectral norm of the inverse of Xν by the diameter of the simplex, leading to the quantity diam(Sν) Xν−1. While the above definition with W locally Lipschitz is sufficient to conclude local asymptotic stability of the origin, the following result requires a twice continuously differentiable Lyapunov function in order to obtain certain numerical estimates in the proof.
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